控制工程经典书籍SINGULAR PERTURBATIONMETHODS IN CONTROLANALYSIS AND DESIGNPETAR V KOKOTOVIC力 iversity of lllinoisUrbana, USAHASSAN K KHALILDepartment of Electrical Engineering and Systems ScienceMichigan State UniversitJOHN OREILLYd electrica/ EtUKACADEMIC PRESSHarcourt Brace Jovanovich, PublisherTokyoACADEMIC PRESS INC (LONDON)LTD24/ 28 Oval Road, London NW1 7DXinited Skates Edition Published byACADEMIC PRESS INCPREFACEOrlando, Florida 3288ACADSMIC PRESS INC(L ONDON)LTDAil rights reserved. No part of thisin any form by photostat, microfilm, or any other meansIn this book, control theory merges with singular perturbation techniqueswithout written permission from the publishersto form a two-time-scale methodology for the modeling analysis and designof control systems. Its goal is to simplify the software and hardwareimplement ation of control algorithms, whilc improving their robustnessThe text is intended for control engineers and graduate students whoseek an introduction to singular perturbation methods in control. At thetime, the book aims to provide research workers with sketches ofificant current developments and hints of important future problems inthe areas of robust, adaptive, stochastic and nonlinear control. No previousknowledge of singular perturbation techniques is assumedEver since Prandtl's work at the beginning of this century, singularerturbation techniques have been a traditional tool of fluid dynamicsBritish L ibrary Cataloguing in Publication DataTheir use spread to other areas of mathematical physics and engineeringKokotovic, Petarwhere the samc terminology of“ boundary lavers”and“ inner”and“ outcr"Singular perturbation methods in controlatched asymptotic expansions continued to be used. In control systemsanalysis and desisSN00765392boundary layers arc a characteristic of system two-time-scale behaviorautomatic control-Mathematical. modelsThey appear as initial and terminal "fast transients"of state trajectoriesAl TiEle. lI. Khalil, Hassan K.and rcpts of the system response. HighHIL. O'Reilly, Johnfrequency and low-frequency models of electrical circuits, which have had6298312lsBN0~124176356a long history of their own, are naturally incorporated in the new two-timescale methodologyIn the control literature, the singular perturbation approach of Tikhonov(1952)and Vasil'eva(1963)was first applicd to optimal control and regulator design by Kokotovic and Sannuti(1968)and Sannuti and Kokotovic(196a, b)and, more specifically, to flight-path optimization by Kelley andEdelbaum(1970)and Kelley(la, b). Applicationsclasses of control problems followedte. as evidenced bACADEMIC PRESS INC (LONDON)LTD24/ 28 Oval Road, London NW1 7DXinited Skates Edition Published byACADEMIC PRESS INCPREFACEOrlando, Florida 3288ACADSMIC PRESS INC(L ONDON)LTDAil rights reserved. No part of thisin any form by photostat, microfilm, or any other meansIn this book, control theory merges with singular perturbation techniqueswithout written permission from the publishersto form a two-time-scale methodology for the modeling analysis and designof control systems. Its goal is to simplify the software and hardwareimplement ation of control algorithms, whilc improving their robustnessThe text is intended for control engineers and graduate students whoseek an introduction to singular perturbation methods in control. At thetime, the book aims to provide research workers with sketches ofificant current developments and hints of important future problems inthe areas of robust, adaptive, stochastic and nonlinear control. No previousknowledge of singular perturbation techniques is assumedEver since Prandtl's work at the beginning of this century, singularerturbation techniques have been a traditional tool of fluid dynamicsBritish L ibrary Cataloguing in Publication DataTheir use spread to other areas of mathematical physics and engineeringKokotovic, Petarwhere the samc terminology of“ boundary lavers”and“ inner”and“ outcr"Singular perturbation methods in controlatched asymptotic expansions continued to be used. In control systemsanalysis and desisSN00765392boundary layers arc a characteristic of system two-time-scale behaviorautomatic control-Mathematical. modelsThey appear as initial and terminal "fast transients"of state trajectoriesAl TiEle. lI. Khalil, Hassan K.and rcpts of the system response. HighHIL. O'Reilly, Johnfrequency and low-frequency models of electrical circuits, which have had6298312lsBN0~124176356a long history of their own, are naturally incorporated in the new two-timescale methodologyIn the control literature, the singular perturbation approach of Tikhonov(1952)and Vasil'eva(1963)was first applicd to optimal control and regulator design by Kokotovic and Sannuti(1968)and Sannuti and Kokotovic(196a, b)and, more specifically, to flight-path optimization by Kelley andEdelbaum(1970)and Kelley(la, b). Applicationsclasses of control problems followedte. as evidenced bPREFACEPREFACEImore than 500 references surveyed by Kokotovic (1984)and Saksenamodel is nonrobust in the sense that it may not stabilize the originial systemO' Reilly and Kokotovic (1984). For control enginccrsThe only prerequisite for Chapter 3 is a basic knowledge of linear limeturbation are a means of taking into account neglected high-frequencyvariant control theory such as that contained in Kwakernaak and Sivanphenomena and considering then int a separate fast time-scale. This(1972)or Anderson and Moore(1971)achieved by treating a change in the dynamic order of a system of differcntialSingularly perturbed systems with white-ngise inputs are more complexequations as a paraneter perturbation, which, bcing more abrupt thanbecausc in thc limit thcir fast variables also behave as whitc-nloisc processesa regular perturbation, is called a singular perturbation. The practicalIn optimization problems this leads to unbounded functionals. Chapter 4advantages of such a"paraIneterization'of changes inl model order areshows that a two-tinc-scale ncar-oplirnal design of filters and regulators issignificant, because the order of every rcal dynamic system is highcr thanpossible with major savings in on-line as well as off-line computationsthat of the mocel used to represent the systeReacting Chapter 4 rcquires a familiarity with linear least-squares estimationChapter 1 shows that the parameterization of the model order usingd linear-quadratic-Gaussian optimal control at the level presented inperturbation parameter a to multiply the derivatives of somc of the statcKwakernaak and Sivan(1972)variables is common to most physical systems with small time constantsThe remaining three chapters deal with timc -varying and nonlincarinertias and other physical quantities which can be expressed as multiplesproblems, where the benefits of a separation of time scales are even moreof e. Neglecting these quantities means setting e:0, thus eliminating somepronounced. A perturbation approach is often the most effective, andof the derivatives from the model and hence reducing its ordcr. It is furthersometimes the only way to solvc such problemsshown how a broader class of dynamic systems can be brought to thisTime-varying systems in Chapter 5 differ from their time-invariantstandard"singular perturbation form. A geometric property of this formcounterparts in Chaptcr 2 owing to time variations of the coefficients as ais that it possess a slow manifold which is an equilibrium manifold for thepotential new cause of multiple-time-scale behavior. It is shown, howcverfast phenoinena. Examples in Chapter 1 include models of dc motors,that if the parameter variations are slow compared with the fast dynamicsPID controllers, aircraft, voltage regulators and adaptive systems. TheirLhen the time-scale phenomena reinain qualitatively the same as in timestudy is pursued in subsequent chaptersinvariant systems. The methods of Chapter 5 take advantage of this strucChapter 2 is dedicated to properties of linear time-invariant systems thatlure by introducing a "frozen parameter treatment of the fast system andexhibit two-time-scalc behavior because of the presence of both a group ofshowing when this conceptually appealing approximation is valid. Somesmall eigenvalues and a group of large cigcnvalucs. These systems can befamiliarity with linear time-varying systems, as presented in, for exampleformed into a slow and a fast part, each of which can beratelChen(198a)or Miller and Michel( 1982), is assumed. Results of Chapteranalyzed for stability, controllability and other: system properties. Readers5 are used in both Chapters 6 and 7, which, although closely related, canfamiliar with the elements of linear systems theory (Kailath, 1980; Chenbe studied independently of each other1984) but less accustomed to some of the nonlincar topics in Chapter 1 mayControd problems in Chapter 6 are of the trajectory optimization typechoose Chapter 2 as their point of departureThey are restricted to linite time intervals, and hence must take into accountChapter 3 deals with linear feedback control design for linear timethe fast phenomena at both encls of the trajectory Initial and end conditionsinvariant systems. For state feedback control, design proceeds by way offor these " boundary layer"phenomena must be properly"matched", Chap-an exact decomposition of thhe state-feedback control problem, be it anter G circumvents the matching difficulty by an explicit use of the Ilamcigenvalue assignment problem or an optimal linear regulator problemilLonian property of optimization problems. Thanks to this property theinto separate slow and iast problems. Advantage is thereby taken ot theinitial and cnd layers can be separated by an exact transformation. Chaptersingularly perturbed nature of the problem to design a well-condilioned6 assumes a knowledge of a standard course on optimal control such ascomposite feedback control, the sum of slow and fast controls, which solvesBryson and Ho(1975) or Athans and Falb(1966)the original ill-conditioned control problem to within a specified order-ofNonlinear stability and stabilization problems in Chapter 7 are greatlye accuracy. Conditions are also provided for overcoming the fact thatsimplified by exploitation of the two-time-scale syslem property. A Lyapuunlike state feedback, static output-feedback design based upon a slownov function for a nonlinear singularly perturbed system is constructed viai A sclcctior: from these references is reprinted in a volume published by IEEE Press,a two-Lirne-scalc stability analysis, which is in many respects analogous tocited by Kokotovic and Khalil (1986)Che linear analysis of Chapter 5 because the slow states appear as slowlyPREFACEPREFACEImore than 500 references surveyed by Kokotovic (1984)and Saksenamodel is nonrobust in the sense that it may not stabilize the originial systemO' Reilly and Kokotovic (1984). For control enginccrsThe only prerequisite for Chapter 3 is a basic knowledge of linear limeturbation are a means of taking into account neglected high-frequencyvariant control theory such as that contained in Kwakernaak and Sivanphenomena and considering then int a separate fast time-scale. This(1972)or Anderson and Moore(1971)achieved by treating a change in the dynamic order of a system of differcntialSingularly perturbed systems with white-ngise inputs are more complexequations as a paraneter perturbation, which, bcing more abrupt thanbecausc in thc limit thcir fast variables also behave as whitc-nloisc processesa regular perturbation, is called a singular perturbation. The practicalIn optimization problems this leads to unbounded functionals. Chapter 4advantages of such a"paraIneterization'of changes inl model order areshows that a two-tinc-scale ncar-oplirnal design of filters and regulators issignificant, because the order of every rcal dynamic system is highcr thanpossible with major savings in on-line as well as off-line computationsthat of the mocel used to represent the systeReacting Chapter 4 rcquires a familiarity with linear least-squares estimationChapter 1 shows that the parameterization of the model order usingd linear-quadratic-Gaussian optimal control at the level presented inperturbation parameter a to multiply the derivatives of somc of the statcKwakernaak and Sivan(1972)variables is common to most physical systems with small time constantsThe remaining three chapters deal with timc -varying and nonlincarinertias and other physical quantities which can be expressed as multiplesproblems, where the benefits of a separation of time scales are even moreof e. Neglecting these quantities means setting e:0, thus eliminating somepronounced. A perturbation approach is often the most effective, andof the derivatives from the model and hence reducing its ordcr. It is furthersometimes the only way to solvc such problemsshown how a broader class of dynamic systems can be brought to thisTime-varying systems in Chapter 5 differ from their time-invariantstandard"singular perturbation form. A geometric property of this formcounterparts in Chaptcr 2 owing to time variations of the coefficients as ais that it possess a slow manifold which is an equilibrium manifold for thepotential new cause of multiple-time-scale behavior. It is shown, howcverfast phenoinena. Examples in Chapter 1 include models of dc motors,that if the parameter variations are slow compared with the fast dynamicsPID controllers, aircraft, voltage regulators and adaptive systems. TheirLhen the time-scale phenomena reinain qualitatively the same as in timestudy is pursued in subsequent chaptersinvariant systems. The methods of Chapter 5 take advantage of this strucChapter 2 is dedicated to properties of linear time-invariant systems thatlure by introducing a "frozen parameter treatment of the fast system andexhibit two-time-scalc behavior because of the presence of both a group ofshowing when this conceptually appealing approximation is valid. Somesmall eigenvalues and a group of large cigcnvalucs. These systems can befamiliarity with linear time-varying systems, as presented in, for exampleformed into a slow and a fast part, each of which can beratelChen(198a)or Miller and Michel( 1982), is assumed. Results of Chapteranalyzed for stability, controllability and other: system properties. Readers5 are used in both Chapters 6 and 7, which, although closely related, canfamiliar with the elements of linear systems theory (Kailath, 1980; Chenbe studied independently of each other1984) but less accustomed to some of the nonlincar topics in Chapter 1 mayControd problems in Chapter 6 are of the trajectory optimization typechoose Chapter 2 as their point of departureThey are restricted to linite time intervals, and hence must take into accountChapter 3 deals with linear feedback control design for linear timethe fast phenomena at both encls of the trajectory Initial and end conditionsinvariant systems. For state feedback control, design proceeds by way offor these " boundary layer"phenomena must be properly"matched", Chap-an exact decomposition of thhe state-feedback control problem, be it anter G circumvents the matching difficulty by an explicit use of the Ilamcigenvalue assignment problem or an optimal linear regulator problemilLonian property of optimization problems. Thanks to this property theinto separate slow and iast problems. Advantage is thereby taken ot theinitial and cnd layers can be separated by an exact transformation. Chaptersingularly perturbed nature of the problem to design a well-condilioned6 assumes a knowledge of a standard course on optimal control such ascomposite feedback control, the sum of slow and fast controls, which solvesBryson and Ho(1975) or Athans and Falb(1966)the original ill-conditioned control problem to within a specified order-ofNonlinear stability and stabilization problems in Chapter 7 are greatlye accuracy. Conditions are also provided for overcoming the fact thatsimplified by exploitation of the two-time-scale syslem property. A Lyapuunlike state feedback, static output-feedback design based upon a slownov function for a nonlinear singularly perturbed system is constructed viai A sclcctior: from these references is reprinted in a volume published by IEEE Press,a two-Lirne-scalc stability analysis, which is in many respects analogous tocited by Kokotovic and Khalil (1986)Che linear analysis of Chapter 5 because the slow states appear as slowlyPREFACEvarying parameters in the fast system. A two-stage statc feedback designthe so-called composite control design, is used to obtain stabilizing and nearoptimal feedback controllers. Unlike the finite-time near-optimal control ofChapter 6, the feedback nature of the near-optimal composite control isCONTENTSrequired for stabilization over an infinite time interval. For Chapter 7, anacquaintance with the clements of nonlinear systems analysis such as areontained in Vidyasagar(1978)or Millar and Michel(1982)is assumedThe chapters of the book may be studied sequentially or in a number ofother ways. For example, readers interested in stochastic control wouldconcentrate on Chapter 4 after familiarizing themselves with the contentsof Chapter 3 on linear feedback control. Chapter 5 on time-varying systemscould be read immediately after Chapter 2, Other possibilities are thatChapter 6 on optimal control and Chapter 7 on nonlinear systems could beproceeded to immediately after Chapter 1 and Chapter 5AcMay 1986P.Ⅴ, KokolovieH. K KhalilO'Rcilly1 TIME-SCALE MODELANO1.2 The Standard Singular Perttrbaticme-Scale Properties of the Standard model9Case study3.'I: Tvo- Time- Scale pid controlnd Fast manifolds171.5 Construction of Approximate ModelsCase studies in scalCase Study 7.1: D￡ in the dc-moteStudy 7.3: State Scaling in a Voltage Rcgulat1.8 Exercises19 No:es and References2 LINEAR TIME INVARIANT SYSTEMS492.4 The blcnck-Diagonal Form: FPREFACEvarying parameters in the fast system. A two-stage statc feedback designthe so-called composite control design, is used to obtain stabilizing and nearoptimal feedback controllers. Unlike the finite-time near-optimal control ofChapter 6, the feedback nature of the near-optimal composite control isCONTENTSrequired for stabilization over an infinite time interval. For Chapter 7, anacquaintance with the clements of nonlinear systems analysis such as areontained in Vidyasagar(1978)or Millar and Michel(1982)is assumedThe chapters of the book may be studied sequentially or in a number ofother ways. For example, readers interested in stochastic control wouldconcentrate on Chapter 4 after familiarizing themselves with the contentsof Chapter 3 on linear feedback control. Chapter 5 on time-varying systemscould be read immediately after Chapter 2, Other possibilities are thatChapter 6 on optimal control and Chapter 7 on nonlinear systems could beproceeded to immediately after Chapter 1 and Chapter 5AcMay 1986P.Ⅴ, KokolovieH. K KhalilO'Rcilly1 TIME-SCALE MODELANO1.2 The Standard Singular Perttrbaticme-Scale Properties of the Standard model9Case study3.'I: Tvo- Time- Scale pid controlnd Fast manifolds171.5 Construction of Approximate ModelsCase studies in scalCase Study 7.1: D￡ in the dc-moteStudy 7.3: State Scaling in a Voltage Rcgulat1.8 Exercises19 No:es and References2 LINEAR TIME INVARIANT SYSTEMS492.4 The blcnck-Diagonal Form: FCONTENCONTENTS3 LINEAR FEEDBACK CONTROL7 NONLINEAR SYSTEMS3.2 Compositc Statc-Fccdhack Contro7.2 Stability Analysis: AuLuntonous Systes3.3Ei7.3 Case Study: Stability of a Synchronous Machin3.4 Ncar.OpTimal Regulators1107.4 Casc Study: Robustness of aa Adaptive Systenl3.5 A Curreileu Linear-Quadratic DcsignStability Analysis: Noromous System3.5 High-Gain Feedback7.6 CumpusiLe Feedback Control7 Robust Output-Feedback Design7.7 Near-Optimal Feedback Desig3. 8 Exercis7.83333. 9 Notes and reference1559 Notes and ReferencesReferences394 STOCHASTIC LINEAR FILTERING AND CONTROL4.1 InTroduction4.2 Slow Fast Decomposition in the Presence of white-Noise Inputsreferences added in proof4.3 The Steady-State Kalman-Bucy Iilte164.4 The Steady-State LQG Controller44. 5 An Aircraft Autopilot Casc: StudyAppendix A Appro ximatiort of singularly perturbed systems driven by white noise4.6 Corrected L QG Design and the Choice of the Decoupling Transformation8 Fcles and references5 LINEAR TIVIE-YARYENG SYSTEMS1 Introit25.3 Decoupling Transformation5. 4 Uniform Asymptotic Stability5.5 Stability of a Linear Adaptive System5.6St5.8 observability9 E5.10N24S OPTIMAL CONTROL6.1 Introduction6.2 Boundary Layers in Optimal Control6.3 Thc Rcduccd problcm6. 4 Near-Optimal Lincar Control6.5 Nonlinear and Constrained Control6. 6 Cheap Control and Singular A6. NO